"Blackberr​y message memory is full!" when it's not - please help!

Hello,
I wonder if anyone could help me please. I have a Curve 9360 (OS 7.1.0.336) that I've only owned for a few months. About a week ago, out of the blue, I started to get the message: "Blackberry message memory is full! Please delete some sms messages". I know the memory isn't full for three reasons -
Firstly because I have plenty of memory left on the phone;
Secondly because I can send as many messages as I want and make them as long as I want - so I am actually increasing the overall amount of messages. The message only pops up when I try to receive a message, and only by deleting received messages will it let another one through;
Thirdly because I can, for example, delete a text message that is 1 message long (in the first 160 characters) and receive one that is 7 messages long (the limit for blackberry sms). So in this sense I am also actually using more memory.
This seems like a glitch to me, but I have no idea how/why it arose, or any idea how to fix it.
I have tried a battery pull, which didn't do anything.
Thank you very much in advance for any help!

Hi John_Clark,
Thank you for your reply and my apologies for the delay in responding.
Here are the database sizes:
Database Sizes
Total Size: 20855.9K
Address Book - Default: 28/11.0K
Application Permissions: 1/0.1K
Attachment Data: 0/0.0K
Attachment Data - Calendar: 0/0.0K
Browser Bookmarks: 8/2.8K
Browser Cache: 1/6.6K
Browser Channels: 0/0.0K
Browser Folders: 2/0.1K
Browser Messages: 0/0.0K
Calendar - BlackBerry Calendar: 20/10.8K
Calendar - Default: 20/10.8K
Categories: 2/0.2K
Content Store: 116/5354.5K
Enterprise Configuration: 0/0.0K
External Bookmarks: 0/0.0K
File Explorer Network Favorites: 0/0.0K
File Explorer Network History: 0/0.0K
FM Radio Presets: 0/0.0K
Folders: 0/0.0K
Memos: 17/17.5K
Messages: 0/0.0K
MMS Messages: 53/9239.3K
Phone Call Logs: 11/3.5K
Phone History: 10/2.5K
PIN Messages: 0/0.0K
RMS Databases: 5/2.3K
Saved Email Messages: 0/0.0K
Searches: 7/1.7K
SMS Messages: 10619/6177.6K
Tasks: 1/0.8K
Text Messages: 14/12.6K
WAP Push Messages: 0/0.0K
Web Search Engines: 0/0.0K
Word Substitution: 3/0.1K
The SMS message number was at 9976 when this warning first started appearing, which confirms what I was saying about this not being a lack of memory issue.
Prince2012, thank you for your welcome and replying. I had already checked that "Leave messages on SIM card" was unticked. And yes, deleting unwanted messages is how I'm getting by at the moment, but it's not solving the problem.

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